We report a fourth-degree polynomial that parametrizes analytically all period-4 orbits of the Hénon map and use it to investigate arithmetical signatures of the symbolic coding for this prototypical multidimensional system. A discontinuity in the symbolic dynamics observed by Hansen while following numerically a period-6 orbit along a closed loop in parameter space is shown to exist already for period 4. We obtain an analytical expression for the locus of all such discontinuities in parameter space and explain their origin. Our analytical results allow the accurate location of all discontinuities, in contrast with topological methods based on homoclinic tangencies that exist over continuous intervals.